114
3 Forced Vibration of Single Degree of Freedom System
in Fig. 3.37b. The spectra for maximum relative displacement and maximum absolute
acceleration are to be obtained.
Equation (3.143) is rewritten as
z = −
¨
x s0
p 2 p
t
0
f a (τ ) sin p (t − τ ) dτ
(3.144)
or
z = −
¨
x s0
p 2 (DLF) a
Therefore,
| z max | = −
¨
x s0
p 2 (DLF) a, max
(3.145)
It can easily be shown that maximum relative displacement z max and maximum
absolute acceleration ¨
x s0 are directly related. At the instant of time when the relative
displacement is maximum, absolute acceleration is also maximum.
Therefore, from Eq. (3.134)
m ¨
x max + kz max = 0
(3.146)
or
¨
x max = −p
2 z max
(3.147)
Substituting the value of z max from Eq. (3.145) into Eq. (3.147), we get
| ¨
x max | = ¨
x s0 (DLF) a, max
(3.148)
Based on the ground acceleration of Fig. 3.37a
f a (t) = 1 −
t
t d
for t ≤ t d
f a (t) = 0
for t > t d
⎫
⎪ ⎬
⎪ ⎭
(3.149)
Therefore,
(DLF) a = p
t
0
1 −
τ
t d
sin p (t − τ ) dτ for t ≤ t d
(3.150)